The influence of foundation coupling on the dynamic response of simple structures.

THE MnT-LUENCF C! : FOUNDATION COUPLING
ON THE DYNAMIC RESPONSE OF
SIMPLE STRUCTURES

: i GORDON HAMMER

Li L

U. S. Naval Lost^niiimUe School
Monlere) , California

im

t r

THE INFLUENCE OF FOUNDATION COUPLING ON THE
DYNAMIC RESPONSE OF SIMPLE STRUCTURES

BY

JOHN GORDON HAMMER

V

B.S., United States Naval Academy, 1944
B.C.E., Rensselaer Polytechnic Institute, 1947
M.C.E., Rensselaer Polytechnic Institute, 1948

THESIS

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR TftE DEGREE OF DOCTOR OF PHILOSOPHY IN CIVIL ENGINEERING
IN THE GRADUATE COLLEGE OF THE
UNIVERSITY OF ILLINOIS, 1964

URBANA, ILLINOIS

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1.1 General Ctcrtc ©«t of IVoblab . . X

1*2 turpoao . • •»••• 1

1*3 lilted of i ^y i u £

II. :wU!fcwl'. .. uc

.1 lot JLng l\s»ctio« . . 3

2.2 .truottrr Gofteldor«fd 4

2.3 i«IX r hjRractcrlsticB esd Interaction with - trwtwra . 4

2.3.X Vertical fexlrllitj .... 5

2.3.2 HorlaMUkX •laxitilltj *

.at

in* irn r,M*Tt'ux

7*

3.1 imljKfai of ^Lgi'- Itectaugular GirucVar c .cdX ua—

yielil^r is Vrttoal ilraction 8

3.1.1 I**k 1 i. ;Jl«a at cVr at ***

3.X.1.1 ^o#3« X, v- • tmnlis * < 11*13.-. ... 1

3.1 .1.2 Zone 2* *lldi«v f to O a arb a wift — ... 15

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» cil -vloHin* La Vgrtito O'xwtftori o*4

lfrU«*J *t Coster of *r»

3*2.1 rt UdLag and So CwjrUinal*^ *7

3*2*2 A^rocbite Distance . true tors -214® . * 26

3.3 fc*X?e£s of ?i ^ JM*t®ngu2ar irueturr oc oil ... >w

Holding fcrW«02y, Umi. i ,U®* *t wrtcr of

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3*3*1 Fhoar 1 •j*®e«l£*i of O^ct la* V. » j*r * . 32

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3*3*3. «*»•» J nation • to . 36

*££*1 Crowd

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Load -*»»• terf • tt- • • . 7

•Igfr' < troctw- 7

nmrihZ t raeirtr* .............. ...7

Arrttc*L ci «f«M* Ity 7

AtrucUa- .t .tractor* i i a*,- 41

far *-«/ Coawftl' » 1 »* . * 2 wv«« to I f ^ . . . - 42

f*>Isfc of li «ti— of Load . Li

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IV. c-4 iEfliwace ef . tanxrturo » Tint ©f ••••••

fisali* * tie* •• V. ^tm> of uv< *>jcWi j-9Wt&

■ rc* i- * V tracw «a -*i ^ •

fir, j^TPCud «te • •■ i ^teUt r « •***•' tar* * ... 55

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TlaUlv >»fc

2be urlt&r ulsfces to axproes his sincere aijrf«istian to
*xctor . .. a— , »— i- * r^f 3*0*0 of structural iegiaoerJ • ,
fur ■ 'f critieJjrs, ear-wi.-u.po.prt, .a * ,•■*!• ■•. »*

1

i*i jhmq& • ~i &£ ^

olaftlaC .r**fert&oe »f a ‘trcfc^ am lRtfsanroeKi ft.- ti» interact! •*-
Ur-mn 14m .true'ar- •*-" ■ its

«ft*a a «jn— 1r load ts^-on a structure* the fni^ a/ ia* i<-** i
esuaaa the structure to neve extaraally car Jefoca leteraally until the
mmatcr isqperte' to it ia used wp* or until the structure fella* 71* in-
turml rest, taooe offered by the structure deeads upco t Im arftr- ^ s»-

rigidity of joints nod ecnnectima as well ee <*f iwlitldual orsuftor%>* mm
a rtrwstura eea a>*- enetr. defcraetin or pflocifo®* no atm-'t rn
hm a Halt to th©~«sount of energy it em absorb without Mta*. ftai
Staa a etractur® Sea* not fail ky Intense! ueSarvetlos, it a^r be
artx^r-eto*7 if it J' i r >%• J bodily. Therefor , a • *-Mf •*

arteml toaiane la alao of t«#y«rtanee»

l* v <tg ooe

> je psrynmm of this vSa**irtatift > ia to fcucy the wasr ia Udafc
a B?=i -> etruetco e «r -:r wiottl c lti«*e ©f eL**l» ..v>a • - —

e««.'Xtio»M of -atiosn reeta^Xrt.

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M

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i*3 rgtixxi v£ i

The Method of exprootih to the problem Is analytical. The con-
trolled paraoeters are discussed in Section 21* fcocticm HI ccntoins the
equations and solutions as veil as tha senuaptione made, section IV cos-
tal .. the euBtscry and conclusions; Appendix C contains graphs of acme of

the results.

Each eyaool is defined whore first used; and for convenience* a
srjBnery of all symbols is given in Appendix A.

3

n. tassum, DExrr^iw of vt&iwu£> ard fiRmz&m

The parts»ters of tho jroblaa fall into three general groups s
thorn character! etic of tho loading functiorsj thorn characteristic of tho
structure; those characteristic of the scdl and its interaction with the
foundation .

2.1 3J» I/aading EuncUon

Tho load, ie a result of a lateral tine-dependent pressure acting
on the structure. The loed le re its sooted by the force f ^ ufdch ie a con-
centrated force, dependent cm tine, and eiweys oqulyoli wit to the resultant
of the pressures at any instant* The point of application of the equiv-

1—1 r x 1. control ccort«t during m? glwn dyu.de loediug. n»
point of application is one of the peraaetere studied »

CfcwiccBly there are a groat nrnbor of shapes cf load— tirco curves
uhicJi could bo used*. The choice ia a ccKprordLaa bntucon accuracy end ex-

•.it

pedlency* 'Groover, too much accuracy would bo itowarranted in view of tho
epproxirate nature of other quantities entering tl» problem.

For the pisrpoe© of this study, it le cienirablo to standardise the
shape of the load-tine curve so teat tho effects cf changing other pero-
swtar* can bo noted* It la also desirable to chocee a ehapo which will not
unduly complicate tho calculations • Therefore, e rectangular pula© chape
tilth varying P^ and varying t (duration cf pulse) hen been used* It is
shown in Fig* 2-1.

«fl %

I V • t »

m *

So direct ooed&oretittD la glvasi the effect of vertical
jV < .» ■— on tie etructu re* wever, sdnoe the vertlcsi treasure <u>ce

not el 0 as rapidly a* the latere! paresrwra, it can be asscnad ti^t
the vert cal .rear are ia oonetoct tor the entire tir» that t ha structure
la considered* T! e *»lgM (ag) of the structure can be * « eufflci* ::tly
large to ttxlxsuo the vertical pressure force*

2*2 * jrjotTJro Coi^Lirro

The ei' le structure studied la a rectangular boa-ahapeU one*
Its mm am be e.---ii'et«d cocoactretod at a center of rsass (c •».) v*3
it ia assaaod that the location of the e.~. ie towns, TJa *e£#t cf the
structure ia ng* The height of the e*n. ia represented by £ • D ia not
raceaeerXly half the height of the aiewsture* T&e uidth of the beat ia
- . (’5o affects in the tfclrc. A5 <■ naion are coceidered and the dimension

In that direction can be thoug t of aa unity) .

In the calculations for ov- r fr a m ings the ratio of ia

A

'V

ir portent* This i« ti keaa as 4 in ease of the teazle calculations ao that
the effect* of varying other part- tors can be studied.

In the first part of the ataC the structure is consisSvred to
be aryielding or rigid as shows in fig* '-*2* It la Irtrr coneidered to
have ra » abllit; to dietort as a "shear ■■ » shuun in Pi t :. - .

2*3 ball Characteristics 1st* recti « ; lt * > trutrtigg

The datemimtiOR and predlst&c* of soil naae propertls* •**

,ng static lea 1 iiv*

fei* t *•

• U m

properties are equally cco plsx* and little infonaotlon is nvailabla »
Purthenaore, the soil boss properties vary for different localitf.en, and

5 *

even vary in the eme locality at different tines* Cksnoequently, in an
analytical study such as thl©, which is both theoretical and appreciate,
the properties of the soil are extremely ovor-einplified .

The soil naas will resist the foundation thrust with forces
which can be resolved into hcodsantal and vertical components* It is con-
venient, then* to define vertical soil dynamic properties and hcrisontal
toil dynandLc jarepertieo*

2 . 3.1 mMsai ns^mz

The soil beneath the foundation ia ro presented by two equiv-
alent vertical springs. The springs have a load-displacement relation
which is intended to approsinKto a eesd-elaatic soil. This is shown in
Fig* 3-4. The value Ky is the slope of the force-displaceescnt curve; cr
in other words, th* force eccert&vi by the spring per unit spring ccopreseion*
It is aasizaed that -'ll# airings cannot exert a tensile farce (pull down on
the structure ).

Ky i# * parameter Uiich can very uldely* It is further de-
fined as ng/2Sf where a is the ness of the structure , g is the acceler-
ation of gravity, and s is tho distance that the two springs would each
compress under tho weig' t (eg) cf the structure* Therefore, S can be
thought cf as a theoretical Initial settlement of the structure which
recurs as the structure is placed on the springs*

m4

u

• I

mmmum

• •t t

6 .

If the structure rests c® the grower surface, i* • no emtm Im
and m Initial settlaotmt, the horizontal resistance is largely friction
between soil sad structure* ‘She ^efficient of rssletonoe is Osei^ssted
as the constant paMNster f# sasimd isiapewteat of tine test velocity*

The wrij^Lt of the ©tractors? is s*C5 the hcriBorrtel resistive fore* Is agf.

If the struct ore rests with its faurjSstion below the ground susw
fees, other hcris -ntal resistive forces act* sheering resistance of the
•oil* ooheeive forces between soil and fewnuatior., friction forces.

the nature of the fSrictice forms is oneertsis* The plmm cf
sll/- sgs i» uncertain. lha cohesive force® set® ^certain*

the vspi^s* redative forces all add to odae the total racists;**,
to horisontel jaavess wot of the structure* the relation of total resistance
to bsrlsoctsl dieplaeseaent is a function which could have a wide variety
of shape* when plotted. It is therefore neoescury to mSm m arMtmry

tecordingly, the total horisnmtal resistive fore© Is aetnsaed to
be constant and independsnt of dlapl««aaat fled velocity of aovenexit* It
is sfasraeteriJBeU ty a di^anaicnloa® coefficient of resistance* • • vrloe
of f is not United to values less then unity.

The value f is therefore used to indicate the horisontel re-
sistance rogaordlacc of whet ijtr the atruoture roots on the croead

rests In an crxoawllon

8 .

II, AJULC*. ..DC, 2 - fi’JSTL •-

3.1 Ana?: res g£ Mrfd ffwetmyc^ar , troctare si all aKsrloidlag
fort Inal Clrttfl'i.,

;.i.i

Ijocter this hMKHnf eeresrel c'iffarcttt eases are considered. They
differ In the acriaption of ths nature of tho bcriaoctal resdctiv® force
exerted on tbs structure fcy tbs ground*

Since the ground is assured to be uanyicMing in the rertie&l
direction , there can be do dejroeeian of the structure into tho grousd.

The only poeeitla action* of the structure ere* (1) cerrturring (.) slid*
ing (3) eliding end crer turning.

Flares >4 throng 3-3 shew the {ptxurel conditions for tide
{rot&sn. fbe structure ita«lf , fig. Hi ia rid- tm*raeA to reel. t all
internal deferretlcn. It la mbit: to elide cr overturn. It cannot dopreea

.ut

into the ground.

TS» co^ihstac x and y designate the absolute disuSaoanect o*
the oaert r of aaaa of the structure, the coordinate® x and y dod.-jset# tho
ataolivte Hepiso rt of the forward toe (A) 'l«e. thr to on the aide away
from that struck Ly /^). it* »•*,!« «* ie the •*■ .ouare of the cloedorlae
rotation of the structure free the aero ;retltlcn.

finca tlie loading is dynontc, the aeoelarfftioa of the aaaa of the
structure tdl- be an tefertant factor, v cange in position of the
structure can be 'oacrl^—' bf v * c.«ar» in x, y, *■* (or the chsn^o in
r, r ,*« ). be ness of the structure, cocsit «red ccnoortrsted at the occtt-r

m M

s** *

i M

9

of aese, can mow with any cr all of the x, y and € dieplaoenonte • There
vdll bo inertia forces resisting these aovomnts, nawalyi rat, ey, and ^Sa •
where :

» - naaa of the structure

x= d * y /dt-

5 =

e- d ' a /dt'

squared radius of gyration of the structure about its cs«

«•

Za Pig • 3-5 those forces (mxjmy,^m6) are shows included in the force
diagraa. In the equations to follow* t' is principle of D’Alcstbert is used.

The equivalent blent force, P^, is considered to oct horizontally
an the center of mas. The farcing function use shewn in fig. 2-1*

Tho horiaontal roe istlva force caused by friction and by other
forces between the foundation end roll is ae&ned constant. It la ocual to
iff I vImto a is tla^ uses of tI>o structura, g is the acceleration of gravity*
and f Is the coefficient of resistance between soil and foundation. T2»

.-re

value of f is not necessarily United to values lees than unity, eo would
be expected for a coefficient of friction cely.

The Magnitude of the horizontal resistance force (ngf) will in-
flncnoo the Manner In which a given structure responds to a given load* If
f la sufficiently large, the force agf will prevent sliding of the structure
regardless of whether or not the structuro rotates. The value of f which
la Juat large enough to do this is designated aa fj. On tho other hand, if
the force agf is snail, there will be a value of f bolov which sliding
occurs with no rotating* Rotation is prevented by the restoring aments

/

Kin rj r( t xV)

W

It 1

t w \ m mHV

•4 - ^ rf

*■ j^IM d ^

B

10

caused by tbs of tb» structure and by tin horizontal Inertia farce.

value of f la deeigneitK m

1) iJetanidnation of an easja-ecetaa for f x

free* fig* 3-3 we hfcVB t!"xs following aquation# of faction#

r h«o

P x - m(£ +D6) * £ M

• •

or ^ " *tiI38 — X = 0 Since no ihdlfng

2 1V]= c> obout -front toe A.

f?(D+B6) - mo^B-ce) - o%n 0 * o

where ^ ^ ; s the Sc^^Bred KqoIrttS o'F ^yfairCon
X R£> + m 9 D o_ 1 RD- rngB abou.t A

Q * 9 ^r

from which

e . RD-m 3 B A | r~Rl4.m fl C - .

° RB+mgD (, cosh l 1 )

.Since at "t = o, 0^0 = 0

£lflbre^ti«Wng twice with respect to tir»$

S _ RD - nngB
® ^ m

/ r N f P.B+r"9t)' .

(« S M * )

R H = f?-mD8 = f3 - mD( R ^'^ 3B )(cosC( B Si^®t)

*£**•» H la the bc^ercotol force mctwmry to keep t&c to© tren

eliding.

ever tf-afc R is e function of t» end tiiat the aaxlaab value

wb#a the tara

la a xdniis&u

It la

Hi

H A ~ V 9 X \ ~ ff

» — rilnl ~ fi

pn ’ * ~ \'i- cr ojfilw 0 = X ^

/■\ r . r : )' iro^f ii od p <2~iV' 1

C — n V - d 'i ~ \s ^ 1 + -I ^

rftidi.'T." *St .Jitn i t r, -«T e; 9 -••orV,

•>/• « /

M d.ue^,x STcf-i - J‘‘J , li •!■ <:I<3 **

n • * £

rn " 9

nj'iritxi fi ~>y t

'a.., tig'f,

('-■»• " *y — I WeiK-^

r. ,rvt ft o^niS'

l

* r.-r l"" OJ n x y

f * *

( mql, yjHgfvi- af? „ _ ** o „ r

i»-;; " lr " * 3 = ** f " "? - «*

f lS'rjL^J!'frizo>'.

\ ' |T ■* 9 /

-f C

11

: ince the -±niauo value of the cosh function is unity, tha mcd?iHic r. re-
quired iei

c 1 ex’ fcerss, which is subtracted ftrca the first, is the effect of rotation.

necessary to prevent slidiigj Is not alwayo lose than unity. Mill® it is
custoeary to think of coefficients of friction ss being Isas than unity, it
siiould be reawabered that tbs f as defined herein is a nos sure of ell the
hcrlaoctal resistance. It nay include ether effects. It aey teko an ex-
tra**? ly large value of f to prevent eliding for oee» values of p/ag and
a/D, but it la still possible, theoretically at least, to have no sliding .

there till be only eliding end no overturning, xt will r* amain aoro if only
sliding occurs.

and the value of f ( required las

It is also apparent fresa equation (3-1) that the value of f

2) Be termination of expression for

ttiloee s certain a ©uni of frictional resistance is obtained.

The equation*? of Boticn fraa Fig. 3-3 ares

rn = o

P, -m(x+D8)*{? H

earl since 8=o

P x - n\ x = R h

._avr<> - irK . -

7 - ,7 “

f^V ✓' X 7 P f*Y

j±/}~ „

o nT 1 '

H
o rr. N

w

*

/« ^ • it d
mm 1! l ^

ftp «

► r

- r/

/? - ( ft cr + x ) r-7 - /?

c* a

r - 7 f v ( o~i +c

o ^ r, i

P ~ y » r * - W

* , *V7- ,1\ _ -

. I ->

12 .

ZM=0 q bout toe A

(F> - mx )D - mg B - o (e4§ ore ^ero)
-from uo ViTch X — (“T^j ~ 9 ’ i) ) 2 _ &fnc€ ot t B o,

Kcuetliig end solving for ft ;

X = O
• •

X = O

R - f?« =

- ^ P

m

Rk = m 3 b /d

But Rh ~ hnc^f*^

'fa = 3 /D

(> 3 )

73b* value of fp Is seen to to intis pendant of the leal parte actors
sad dependent only ujew the geajatrlo iropoartlooo of th> etroetero. “t la
to be vesnafcored, however, that tl» above values of f anti f era obtains!
* -mates tte(t tto ^ulTOtert tore acta at to. ....

Th. m-to for f , and f 2 ara both .tad** Item, for my
given value of B/B." T3» f^ line* Is the bounlary of s aoee of cwrtwrainj
only. The f 2 line Jb % a boundary of a son© of sliding only* As shewn in
Fig. C-l the two lines intorsoot when they nr® plotted against value* of
lead. This forma four sanest (1) Overtsnuing Only; (2) wilding Gtdy;

(3) OverUr r i djag and Sliding; (4) Ho sliding and Ho Over tom lag.
lha throe acres involving eztoroal aovunsit will now to considered.

3.1 .1.1 JjBl i PvnrtiTtiing; . .IMInc

If there la no sliding of the structure and coneeruortly no nove-
oant of the front too A# tie only poeei- 2o notion ie ovorturr'ing about

. L

/“* Z'f-'j

ji..c do G - ,Y! 3

j'u ° f- tO r S gen - Ha Am ~ ,-Jj

— r -v

,o = ^ ic Jj jrt'iZ

C>= X

• »

0-7

3 -> _

S' A -

f~J > _ 7 ".

m j - *

t'l ^ t p PT^ c "T

c * 4 - vi. •= k^lg

Q rn /fv

qa 1 e ,v r ‘

^ t ; m 1 u

.

15

y era me . t*rv*fan. tef*» lag to fl*. >2 «rit'»^ t* •

♦cuu Men* «sf rrti— f or t*. a <* of tfca a oat * » c^.*

( p, - rn X X© + Be)=(rnQ - m y^B-DS) + ©

-for O ±t ±t r

(. m^B + Be)=(m 3 -m3XB-:DS)+ <(m 6 ■’>>

•f o r "t p •< t

thi ©full e - -** H’sda ues t&n Mall «». > «r •

«*ala # sni 1-006 .Ma would introduce m orror if ♦ IMM 1 >V •
hr a *tructar» .!* F/D c •->-< a«rl mla wl ^ be about *- * °

(iu i© • at of irwta-‘lity for •awortcralag)* for thla »• ;b, € diX'w'
tr*M t u « ty about 4 s poreact _i*. 1 differs trcm .a # fcgr eboet 1 ? par-
lor* . r 2 »a« errors wltMa the accwrecy 4 * of the ether pm<(o
^>r rtrueturoo with /i then . the < on $ to b» owolt with la -***
tteoi «so tv* «srr.r* vll. bo "*•««•

.A

Her slrwai r' » it* a largo /. ^ 'WnU* la not innoMl U tha

Iryvtat t *^« 1 ori ‘ii.

»• urn r *. «c-att«a. 03 ) u». 04 ) uiM Me*- r «*

»* 1 * tfcla cawo, wo car write:

X •= £> ©

y = - a e ► of ^

e = e

&««t-Ut 9 IrtO ^ VO

(P^ - m ID© Xl> + B0) = (mo> -v-m "5 ©X"B-I>©)-f ^ m 6

•*% '*

«\ ■> — j ~ TO )*

~i ) -~ v 9 r y >j,\

G rn^ 4- ( ^ 1-1 lj[ ” c - f^ri ^ d.-f cr < x rr. ~)

it > r* J *107

■HM JMP Mi •#* - — • * * - " ■■ I

3 /
^.£J -

4

X

L*

Jk

o*'-t 1- v ^ -i ~ Si & >i ■+ ~ (> 5 + J X G 1 r.i - R )

14

P ( D + B F? © - m D 0 - ^ g B - 0 + <"0 E^0 + ^ V* 9

(^V B*+ D^jvn 0 - (mg!)* P,b) 6 -f?D -mgB

03.1)

Slnoo (B 2 ♦ D 2 ) ie tJhe r*n^re<2 distance &m the c. a* to the toe olxxrt

whioh rotation is taking place* <? *" ♦ B 2 ♦ El 2 ) is tK^iivelent to the square.

2

rsUitw of ©-ration of tbs nose about the too (fiosigayated ty ^ )• E q-

uctlcn (5-3*1) can then bo urlttom

^ m 8-(m3D+ FfBjG * F>D - mgB

e- ^v^ 6 ^ »- R °~C 3B

cr

and in a si liar sauner* equation 04) will boca • t

<•%

^ * w 0 - (Vn 9 9 - - m^B

03*2)

or

3B

(3-4*1)

sene aquations* of coizruo* could hew toon obtained frcn Pi*. -2.
If the structure is originally at vast sna has no initial In-
clination 9 (l*a. e 0 =°, be --ff = ° }. 3he solution to (>3*1) is»

t -O

e » ( B/p - ^"1^ 1 (,. roih ]^^lfr i )

(I + */m 3 B /p ) ^ m

and the angular diaplacataont and angular velocity at tloe t ( m of pulse)

e P ^ S/p : g/m ! ^ ft- cosh. _

0- +R /«3 ®4,r m

1 1 - •*.

+ (?B

p /

Ui >«K.

t'* 9 -h ^ -3 r - - W Q - <5 fl £1 r Cl R

a^n- - aV’^vi? +7rr.O" ^ -»"%)

/

H . 1 " • C'rl 5 9 (d } f) -f- d , ^ cr*

y.wr. -a_rf

n •*’ T

a ! j *•

- - 0 , X

^ (

\ „

J

?

J

9 * ' J c , . » f

- - ^ i n<? V)

/<• ^

^ = 0 h S *

n r

o- *

O -

c-

y

X J
\ 4'

.9

P In’

Y

r- j

- ! r.'

5j

1

/

X -l

y

15 .

mgD+ F?E>

t

?

The solution to (3-4*1) is:

= A cosh

+ B Sin

+

uliesre is Measured front poaitlce © end t Q in awsaared frc« the tins the
pulse stops.

a = e P - s /c

a-

The position of the structure when it etope overturning# providing it

doesn’t reach the point of overturning instability* can to found ly fitting

m

the tin* after the poise stops for © e to equal aero* and adding tl» ti-» of
the pulse* Then the corresponding angular diaplaseaeat €» a83 . can to found*
This has teen dene far s masher of different value# of pula© farce
and duration# and a fatally of curves was obtained showing the aextiim ©

.A

which tba fffcrueture^ would asmas before rocking beck* F«r ecy structure
there is «no value of ^ which will be critical for instability# sod the
structure would then continue to overturn under its weight* These re-
sults ere shown graphically in Fig. C-2 of appendix C.

3.i*i.2 m&Z tJJmm su* -2

FTna Fig. 3-2 the eouatico of action let

FJ =• mjc + mo^f
R - mgf

or X =

rr\

^ f Q,.;r

: ~V ' * r ‘

• ( * t j

2 (r -jjv .

4 , 5 1 ' 4 jy , > ~

u * 1 « - '-5

7 \ l -?• £- * — - H ' « J + ^ i 1 “ i P 20 '-. h - .&

'A"* " o) G 1

T o“/ N

A

•a

Aj PW> |_|- ** J0* « -’ Afc4l W

* A ^

jp -0 ^ , r * £. i -i L* panpili •fl® W V® # fl

% AM A K9

’

4 mh la " 4 » *i|r <|| __ * % rjr»
| <41 fk •* 4C< *4 I • 4 I

p '•♦ ■• ^ <4 (WaM^*p fl

— » fl

.

14 *4 »tJiNa

> £'” J \.'-« ■

i , ■

y . c

Sines $ * at all values of t

end

X = X

.. f? - rrtg f
x ~ no

If the initial diaplgeceont end velocity in the * direction era 0

ft-mgf \ t*

X = (■

) Vs

and the vulocity at mgr tine

^ - y — m9 ^ )*

at the end of tee pulea (at and of t„.)

/R-ma-f ^ t p
*• p *“ \ m '2.

P v m

She kinetic energy las

^ _ _ j_ jtl. r R - "u •*

K.f. - 2 mV - 2 v m / '•p

Since the etructuro is asctcaad to be rigid and to undergo no in-
ternal da f errant! on, this kinetic energy will bo dissipated in triaging the
structure to roat against tba harlaootsl resistance, ragf. 5ho distance the
structure*, vi U clitio after the i«loe has stopped ie titan obti&d d fret., the
energy equations

K.E, of structure at t * External wor after t

P V

vn R-vr>af )*■>. _

— iS ' t P -( x »X m sf)

shore S ie the distance the structure sill aline after the pulae stoce.

8

X “X

+S- -'H

frt

- X

|H(H ft

v j- /

\ V

(T,

) ~ c

- x

- -H

rr

j) +%

*j X / _ r _ Ji ? rrf - i \
~\

rrt > <7

— Cl X

+' 1 h I' - / ’

rr*

- .X

* 4.' JY. r _~ ^ _a'. _* -

Cf 1 \ • y J _ -

r -JT = .3.X

V r > * X j ~ 7

*’ _r£Y ' 'll XL

$ r‘ ->

17

Solving for x #

” 9f v m

E»

of ip nraJ i # .

as* total distance tbo structure tjUJL slide io tie am

X hoax * p 4 * a x

= ^R -mgfvt^y p t \

V m ATaTXrng-f '

cr la dlf'easlonless fora
* ma - =(^

X>

(3-5)

It is seers that the mx±SR» distance* the structure will elide is
a function of the aagnltudo and duration of th» load tad of tb: ooafflclent

of resistance.

3.1.1.3 £&£ i sal ja&gfctg

If f 2< -f < f there is toe such hcrisontal sliding reeLatance for
sliding only tut not enough far ever teaming only* 1!» re salting action
will be s coshlnatiac of sliding end overturning »

Bflferring to Pig. 3-3 end writing equations of notions

*

Or

.n ^ te

-i v

fT * * ^ t

xx r*« ^

>"' ( - ( -f ' I

r, • tt. ; v J "

, .£?_ =
1 T' f l - )■ , • /

, ' a s , .

ix, iL i - L .

v t ^ A ^nr-/ s £ r,i ) **

yi y
—

r <>

>

• 41 haw

• -

2T \V\= O about toe A

(R„+mDe )fD + B6)- rnsCerDe) - ^%n 0 = 0

Solution of the cbovo two simultaneous equations will give values of $
end x for any time t* B» aaseUesa values of € and x would be found by
first finding the x, x, #, and # at (cr*l of pules); then rewriting the
equaticne abovo lecving out the term Solution of these equations
would give x, x, 3, and 6 for any tla* t # (after pule©}* Tbs time fear a
nextaur? t* er X would to found by setting the exproseien for 0 or x= 0,

T!» aaxjUauM # ^ x could then be found* The solution for % la complicated
by tte feet that one tears will contain the product (»K.€8) .

Solution could be obtained by lteuaasfe’a /£ Method; aamning

• • • • X •—

values of 6 $ x far the end of tine increaonte* solving for 6 fx* and sub-
otitutlng the valuos in the equations above* If the equations are satis-
fied* the values correct for thst time*

•«

Fouewr, ff the product (sebli08; in the eqwaticn ebova ia con-

-.•i

• %

eddsrsd to be very eaell capered to the other tarns cartcining and ia
neglected* the aqua ticca boocnei

P x - mi( - mD 6 = Rw

or

Iff- m * = f? H STnce X - Vf D0

and

fmi -C^D+R H B)e=

o~H Z

O - (i fri

H / ^ ft a f ' - X rr* - p

^ -* c J I n c j o ~ i 'I

v'0OS9> r^i - (fctf+a) (? CTr-r f 3

£

• * *

IM !• «a «r

w ? f

H 'Z r ■ - X m

-3

^*.i -i V = A 'jinic ft ?! "- 'X p< ! ~ |1

Ji *-> rV - J_ \ Z. h -' + £ f ri j ~ ? r*' 9

I?. >.0-0 •. •• Ua U hKL J .V-a <*> Mi. (%. .>), mi ^ w S'*

flllf, U« wr.^A ttei |k kc tee H(lll * T( . Ete

•r^Un. «ir l*r L> V--» Mt t'«t f«»** f'Jtr l P»i, (J.j.x) «<U %

«M 4fir» fa: ©v ’ onZnr - W iro i. gpi S re' N <1|. C-2# 4 C

wi »’• for /D « . • • • the )Srt> the aim • . be

etitev - for 1‘j In »£» «2» of the eri-U*^ «

Te vulm of x m he obted»rO r<r* the -. 'J.oot

Irj” nix = R w coi+hx$X=o att-o

r w> » l < p ar Ct > ®*t

-mV - Rh u)itW x=Xp an<j'v = y(p att = tp

., «e *•* ci t> p t i of 32 Cfe. te tbtsJ. eu tte reietl**:

- x = v - d e

■vr • J*-j the feme ift e ->XX*j •*„ tte

-r

«<i> 4 f te i insctu re *

3.1. Lfftet jj£ ; £&& - .V- j* ^-2 - m u --*> ^ ^ Qfld

la the e«’r^» o> to i H at the ■*••►« U*

struct** + This Ifl « r wefc: . eeertyX ^ if the wiW f mb* U . - u~
hte.1 tie *a»(trol4 of tte j«*i»ae ■ .]«s>^ t th CJ-e*. 1. the >f,1tte
•truc'w* "oeilitri •• ter» t *d i eeul .c t ♦<* **c<ne<T of Me • » ,x»* *-»

Vtv* t Nr |HHtri( Mter - rtew ■•*'■•

r-; r i? y pnt

- //r*

-Q*

o i = i .7 x = x fcr j c-X - x Wt ] to //. - V '*'• ~

i

<

T

y

x

20 .

rj^»r , it i« preloads uwt tl*- *»-•» Cl • tsribulloa cou.'< bo »jc*
tfcrt u » <~«ei tlo,>' »»pv r c v obtain *z ana the Ilf-* cf action ' ^

«at^- •! vt« the c«n* It ia of it^.rwt t«ji to estuty l>* «-'fect

of 5-»1a- the joint of *• flloetltr of 1\ with njwit to the c. •

tt» effect of or» lapocrt at the c«* . — to r «» a hcriy** ro-
se ti<» Igr t-% Mdl cn iha fouataticn in the e3*^tloo an***.* to that of
h»wc% vs u.i o». tloo of a^li atic* of the tnpnot e#w*r ni^erv on
tb* ctrocturo it la theevettcr ly poeei>l- to rose a < --dot '*-'J d- r< .-iifte
in aero heriaoetal f *a»* tlon ™». te tics?« : ia point ia ti«* center of or-
ii—l nr of the structure. V t-« ;*>l»t a. ylSo*tion «aqr. -.weed fW:-«»r
L ff * » to direction fcmrk: tl* center of percuestca the barli «’tcl ro-
ttcVm of eeil on the f — f t »aw would be tne seme direction ee that of t»<e
1*7 act.

/Tret, tl# IccsV om of the center of peron^alco id.ll »• ioa» .

I' *t€* 1-4 (•) the fares* acting «s the struct**® jnst rwerte W p ei*%»
ar» abets* — *• ^ act. ft ti- «•..*, ’which is a distance • a tens the c*. • It*
horleontal force mist ti«* be ea* *, fr.;. the definition of «*.
tr»r» util to r. «li i«s£ *sf *•» toe* tv> force ©oueti-eu ere*

Ih=o

But srnce x= o , x= £>6
P, — rnD 0

Z )V1 = O about the c.nn.

(f? cO " ^ Xrr > 0=0

. OS

t g*m

••

e a rr* = ,-i
.mo i'llf JucMj t, O = M Z
- C C fi)

O-

21

Cuhstituting ff= iai the sbovo ec notion

mD0d - ^m0 = o

x>a =

d-

Ft** a seetetngular prists of com tent mss density with height 2D and base
28, 1 1» n^uarcti radius of gyration about cn axis thro- a, its o.tu is*

and

\Z

* si - 4-(^i)

D 3 V D '

^-( 6 % >')

CW)

If tha atructur© ware rod-libe Id shepo ao that the dlraonsdon B war©
negiigihla compared to D, the exrroeclon for d becaraee*

a = i*D

.a

TLis i » tie third point free; the top, and in tha case generally shewn in

a

phyeica textbooks £<s^ the cantor of percussion.

Equation (3-6) establishes a point of a^licction for Pj »> -ich
requires no horizontal resdetance force between ground and structure to
prevent sliding# Whan tins equivalent isspuiae load P, la applied at this
point, the atracturo will toad only to overturn, Th® spproadrjate aeodnua
angle of overturning, w can be found by equating the angular impulse
about the toe to tie initial angular acexmturj of the structure about the
tea. From tins imgular arooentm the initial kinetic energy of rotation can

earn ^

O = cl rrr X y -- b & CL ra
'O = oC r

y

22a

t» found. From the x. B. the oaxLman angular displaoeRseat, can bo

found*

Angular IhjxiIoo cf load » "t p (d + D)
Initial Angular ffeaer.tiBa of Structure * 3^
^netlc Energy of Itotatiou * 4- j (Q\%- -±-

VYl ‘ ^

Wowont » tno ('^s- 'D0'>

( 0 rv,ay ^ r 0 "' a *

0

+©«

m? 3 (er

Work

M «* 6 ’ l 0 W> 9 (B - D 6) d6 = rnq ( b^ ; £ ^ )

Equating angular lapulaa and angular aeecnfesi

_

F?tp(cUX>) - m ^ 0

3 -

m ^

Initial X. X.

fT. t. v 2 . k ' , ^ Q ' 2 m

Equating the initial X* E. to tho work dona m the structure

^t£(d4D)" = ( Be _D e * )

2_m ~J v max 2 >^<xx

Pbr w*dclis

C, - 2 % «», * -S ■§* °

v, - % *iaiMV®v|57

Since at 4> * !?/l> the o.ru of the structure has reached e

■ 4 *4.

position vertically ovar tho front toe* any rotation beyond tfast posit’ on
weald result in testability and overturning. Warn choice of the ♦ cifjn in

tho enat.loo above than would ftive a scan? nglces

since © wo

max

( -
V xx n

G m = G

9 r r , l -f • CZJ

N cr + 1 ,, j ,i

ot

i.

A ‘ - J. „ - - 6 t --

A ? <r ^ ~ \ ', 1 *
' »<*>+

rr • **r<* -

‘r* * rr/ - '3 I-dt}r,rTj* i ~^1:' i A

/ N '- © N

CSa-. 9 ;

,, ^

<». f, ♦ <V ^

— -4 1 ‘ i >

6* yM - {'2 + b' r il?l
i'T+ b) ji'f i „ k

V ■'M

X _W f ^ V 'it ii i fcl+li " -it. t*_\ j£ : q

u «lt

(J

M _

J'Tr

V/M f

. 9 *;

v l I^'b > yV + *l

it! _;

.i —

y

O ^ '

-■ =- £ L i JK r.^] + „,„8 / 3

A

X^f

' / e _-=• ay* d

- ■ a . L m\ y kj \ j \ vlN

f*

23

be greater than & D. Tim equation to be used la then*

0

a /D-T 07 )

Jh obtaining tbs above actuations, aancnte were taken about the
toe of the straeture, If the hcrlecntel reactive force* which acta
through the toe, were not aero (l*e* not applied at the center of per-
suasion)* and if the horizontal rnairttve force were large enough to
prevent any eliding of the rtruetura, the resulting equations would lusve
the asm appearance with one exception--— the distance d would be the fiictanoo
to th. point of .reliction of ^ kmukiwI wrticUj. upcrt Bern ck In-
ctead of the di stance to the c.p* qua t ion (3—7 ) can then be written in

generalization of aquation (3-8) requires that tlw structure does not slide
for any position of P^*

( 3 - 8 )

then P^ ia applied at the center of porous -icn. The

Sene qualitative informtlon can be obtained frees esaadninp

, f v^. n t-i .)*/ \ s ( ± 3 - \ \<Sf _ Q

( isi.) -^~J_ { 1;r , / v ; - i CJ ■ ' ; - a\ - ^

i "a + ri \ » ,

v - p

a

tr ii MM m
%## p|U» 9 I

ii ** %b

i •

*/ mm

ft#

I Ml

r

l(-r'\/ V.CP >i IN * \ff

i *4

J

/'-l ' 1 cb-.'t

\ a > t

gi ^ ♦ %

m U ft V

24 .

alee ftr» aero (noaoured upward frca c .u. ) * tho total ov.«ntity under 'Use

radical will rot smaller. Thia will make the value of € get larger* la

lasct

other won? , the same lapulsive force applioc at successively higher
positions above the c*a* will produce largtr rotational displacements of
the structure*

If the frictional horizontal resistive force la not profit enough
to prevent eliding* it la not ao einplo to predict the zaaxi.gsaa rotational
displacement by the energy method previously used* Fart of the initial
kinetic energy imparted to the structure ia used up in overcoming eliding
resistance* and part ie used up In overcoming rotational resistance. It is
true* however* that whan the point of application of ia at the center of
percussion* all the energy goes into rotating tlse structure* If the point
of application la moved sway from the c.p* in the denar ard direction (to-
ward the c*m. ) sotac of the energy goos into eliding the structure In the
sane direction as f^* and less energy goes into crvcarturnlng* If tho i joint
of application la agyed tip’jrd fms tJte cantor of percussion, sorae of the
energy goes into sliding also* But In t is case tho eliding ia in the
direction (opposite to f^) which increases the rotational displacement.

Iho structure ia than mere suieaptiblo to overiaarning*

Is appendix C-, Figs. C-4 and C-5 chow sono of the relatianahi pe
discussed above.

3*2* tesl Tip flgjfrlt Jiaclaj^idai: aa Skill .fryfomiae JL.

xsSissl UjstslSssxs JkMd, isclku si Qsnisz a£ ixxa

25

In the analysis so far, the structure studied dm assumed to ho
rigid fled unyielding. It dm ae . ivkxI incapable of using up my energy in-
ternally*

It la sou desirable to eccmsine a structure which can yield in-
ternally , and to ? • Kcmt oo orisons. fhe structure ia shown in Fig* 2-f» .
The nec# of the atrocturc ia considered eorscentrsted aa shown at a height
D. The top and bottom hcrlzcntal girder* are assumed Infinitely stiff.

Hie vertical colustr are esauaed to have flexibility. This allows the
structure) to deflect, as shown In Fig* 3-5b, with the to; and bottom girders
ranaining parallel feo celled * cheer b&cm n deflection}* the stiffhooe (k)
of the structure ia tha fcoriaantel dicplacemort (A } of the top girder
relative to the lower girder par unit force acting cm the c*n^ k ia assumed
to be linear. It ia a to speed te ctlffhese including the action of both
eolusna. The whole structure la still free to slide er overturn, ties tho
previour rigid structure, depending on the balance of forces.

The etructur.s is again assured to be resting on hard .-•ourxi, un-
yielding in the vertical direction . Tbe horiaoctel resistive force of the
ground on the structure la again n,-f , where tag la the weight of the atruetue
and f la the coefficient of resistance between the soil one structure .

The equivalent concertrotad blast loading, ia Moused to act
through the center of naea of the structure, ia the 1: puleive load tribee
the structure on a hori sarrts 1 lias through the center of nasc, it if
"filtered* by the Inertia of the mbs* The net or filtered force le *. -sat,

II

where m is the fcrce vdth vdilch the nose resists acceleration » ~jlb

26 .

"filtered* fores will then produco deflection within the structure by peac-
Jnr into the oolwn*. The col-acne deflect elastically or plastically In a
feerlaocttal direction until their deflection (A) tinea their at SJCStmm k)

N

la lar-.c isnaugh to balance the ^filtered* force This force is then

transferred horia rtally firm the base of the structure into the gr c*sad*
Reedl x,s to say, the for os transferred to tbs ground in this Banner cannot
exceed the bcsrinantal resistance (npf ).

The flow cf forces described above my not be able to balance the
load of P^» especially since tbo deflection on tbs structure ia Halted by
the horizontal resiatenc® (xagf) of the ground on the structure* If this is
the cose* the whole structure till tend to slide* This causes additional
novwaest of the aasc, and cotxx*que©tly additdcml inertia fereo to crocrto
the needed balance with ty

The other eoctrene case would ba that of the hariscntal rest ■ tsno©
(ngf ) being adsoastp to prav»rst sliding but with the structure itself r®~
sictl a • further defWsMrtisQ so that the load I is not balanced. In this

rk

9 there vculd be a tendency fbr tie structure to rotate bodily, thus
producing rotary inertia resistance to help balance

Thus, ussier different conditions the flexible structure can roe-
pond in several ways* (1) It can absorb the Whole iKpulso internally with
no sliding and no overturning* (2) It can absorb tbo iapulee by internal
distortion pins sliding* (3) It can absorb the tspulse by internal dis-
tortion and overturning* Tbo waaebaua die laeeacsnte for the different
notions need not occur in phase.

. 17 *

In general the effect of n*>:ihllitj on the etructiiro ia to
rc«*ueo the bard *n placed upon the fotETdsticn.

3«2*i ^giae aai & js^&amag

Ifea too docs not nervo end tho angle 0 rwaatna aero. The struc-
ture corns distort end the lames moms hcrl can tally « distance x. It is de-
eix'od to fin! the limit lag value of f fear this action* Rccia Fig* 3-5b*

The sheer in the eoliweas ia resisted l$r

KX = R h

Z H=o

FJ = rn x + KX
ZlV^O about toe

(F? - tnjf ) D = (B- x )

Substituting

y*

tr*D - m 3 (e>- x)

Solving for *

X (kD + rocp —

mQ B

X = —

KE + nacj

&1 ce K'h

iutv

K m 3 B

rn 3 ^Vr>

\ + "’Vkb

r — Pm _ ~B/x>

' + ™9 /kd

t

r — * • ,i
H / 1 ~ A / i • •

n ~ H S

X / -{• X (TJ -•{
.^Hf r~ ( V;Z

- ( i >->’ n ~ \~1 )

< -d ) f ^ - .1X7)

".7 ; r» ~ t ^ < 1 X ) X

A IP i. , ,

•+ t,i

r £?i \

rT

‘i \

f. ,V

* ' ) » T 4 .

A ^

j-.Y ~ „

r- ‘‘ r i'lx

O'

a.\_A.

r o7 + -

7/l\- '

r' 1

/j ^

~7 - 7

«

.r k t Om * 2 t »» la*. 0 (i*e, tK ■ taraatjure .#*, 1 be rift*' ) tt*

•U** . * wdor .---•- •i F * l/t vfcicb i* -l«t wt» -»»vl •>.

IVr • ff'Tvi , far Uk> 11 ’it of r 'id!** with *0 creertarniu * la

of U: j»» pr .I^W) 'i.» til ir. * • * VrLv*

r d >m m mi£r ill sll t . Ttm

for JMill e Last* a a flections - 'thla t'~ structure.

If the etrirturo *■**«. the ► «h to* ion In a -* '«<

u $*• '■» hoerl*- Htel Zwm* U ’» w to the ground util bo ** .1
W WiJBol'.t to the J-2*ette tf I t Cn^tw-s. - •wthe* - —t

UAs «coMds the **fc» «T toorl«**iau. viHiIwi (&) wt U «t* - ■ t.

tW stun* *ll'- r -• «p not.

J» 1 .J ^ ^rotLiti: .tructag, * 11

t> ’Mtlal C* • lajurted t the dsrujstur i« awr*. «r» In
U ^ wcl 3*e fleetly «r«l la • »; nr. ^reariLaii » Wt «■ f ?

.a

%A» the street i** iUVi eon U «'.Ule>c i> U« tmlm nf

*w

r» 1 »t*.— As t»fe-~#*the X. . . of tft* un of the structo* -a be

oHftlreC by ss-Tri n. tb# IMUt sowurtar "psl to the Ir.ulse o« Un Xo«o

* Neo-art**-.

ff tp = mx

° r * - Rt P

9 * Ud 1 • J X. x “ ^ —

K* E. = ‘/z hn^x) 0 '
d x m

^ rr; —

U |

49

s* I

r r

nA 1 ‘ *

~ * 3 '< l

y* ^ * »

.-fv.i^ - — —

rry

> V-

29 *

The mc> id 11 mow horizontally until the horizontal force in the columns
equals the available horlsontal met stance of the ground on the structure
&*> the entire structure will slide against t.» resistance ngf ts>-
til the balance of the energy is used up* The deflect & structure Bey then
recover its original shape* but the loos of the structure is eeeuaed to
reaein at its oajrfmsa horizontal dlsplscooorit- deferring to Fig* 3-6, the
work necessary to deflect the colossus 1st

Vc = ± jfeaif)

Iteferrinq to Fig* 3-7, the work In sliding the structure 1st

V s = f mq X

Equating initial X* E. to wcrk done internally end externally*

K. F = V c vVf

►dying fcsr x, thr cj stance the structure slides t

- _ JLf f? _\ a _tZ&- _ jcnaf

A 2. \ qna ^ -f 2K

Dividing ly D to neke dlsxsnsionlesst

<> 1 C)

The equation ebovn aJxjus that a noa-rlgld structure will slide a leaser
lats&oa than a rigid one. isrthemore, a given flexible structure on a

given foundation will toad to reduce sliding by the mm mount for t i
values of inpulsive loads.

w tk

_ - V

•'•v V

/

* T ' iX

:v /. a/ =. 3 .;{

. , »

a * i " r V *

/*

S alf

v r. ' > -1

t c i

— .V, — y V _ — i '

T

( , ftv y ^

X

r _ J? q„ T -A \ r _<i

<X*v , v / jr / v <» # a t- J "j

30 .

3*3 lealwis of riciu rtrocturo fa r-o51 Yielding Wrtic^Hy . load

&£a£ k£*Li*a&

Thus far the structures studied hsvo been assumed to be resting m
ground %Mo& wee unyielding vertically. Ac the structure received the 1»-
pulec, the vertical thrurt on the ground wee resisted uithrut any yielding.
Since tide nay not be the case* the effect cf yielding of the ground verti-
cally will now bo considered.

As shown in preceding analyses, the overturning xacvaent of the
structure involves the horizontal resistive force of the ground on the struc-
ture. The BKSW&t opposing ovm*tjcrcing involves pertly the vertical ground
reaction on the front toe A. Accordingly, another limitation is placed on
the stability of the structure; i.e. the ability of the ground to resist s
vertical thrust.

Sbs structure is shown in fig. 2«4« It is supported on two verti-
cal springs, A and <1, each having & linear stlffheca designated ty k^.
k ? is related to th%jropertles cf the structure since it is defined ass

ufeere rag is tha weight of the structure and S is the vertical eetilat, ut in
each spring caused l?y the wight p g,

Pj_ is aseuned to act in s hariscntsl line through the center of
mss of the structure. The structure is assured to be rigid. It is also
s e cure d that there is sufficient horizontal resistance, agf, to prevent
sli- ing. These resaaptioos liait the action of the structure to overturn-

31

ingj and that la the action which la affected primarily by the vertical
yielding of the eoil.

It is also true that vortical yielding of the aoil beneath the
structure will tend to change the beri aortal sliding reaiatance. this
change will be an inert* see generally, and till oaks the previous eliding
calculations a little safer* T e suitability of the previous overturning
calculations, however, 087 be affected; and to study that possibility is
the purjOse of this section*

Inferring again to Fig* 2-4* the springs are seamed to have
linear stiffnesses as shewn* In addition, the springs are not able to pull
down on tbs structure, so if the structure tends to lift off a spring, thexv
vg.ll bo no restraint.

Deferring to fig* 3-0, the action of this idealised e true ter
under an iiipulae load is as follows. She forward toe (A) will depress
spring a and take increasing vertical thrust free the tfrouna* Tt*e rear toe
will still rest on rjring C, but there la a tendency for the rear toe to
lift* The center of rasas of tbs structure will be moving downward.. Finally,
the vertical thrust under too A will reach a value equal to the «• ►hreie
am of the weight of tbs structure and the inertia force caused by the
vertical novation t of the nose* At this point there will be aero force dour-
er! thro ;h toe C, and toe C will be froo to lift* Qvertarcinf will then
start to t«3ca place about too A, which is in a doproeaed position.

When overturning starts about toe A, the o -ntor of aasa will re-
verse its vertical oa * xanort of novonetib and nova Howard* This oeusea a

32

te* A further. But if toe A depresses* the center of msa will lower too*
This tends to counteract the upward Bcrvanent of the o*q« caused by the
ovtnrteralng*

sparing were relatively soft* the result would be for the Base to rotate
without vortical noveoest of lt« c.e. C-xing A would dejare so the ssiount
reoeaoar? to permit this*

If the oats of the structure were relatively snail and the spring
ware relatively stiff* tie result would bo for spring A to bold the position
It had when toe C lifted, end for the structure to rotate about toe A*

The action described above any stop at szsj point of its progress,
depending upon the prevailing values of tbs problem parsr ter a.

Tlio action can be thought of as being in two p basest (1) re-
pression of front toe with rear toe not lifted. (2) Overturning about

..a.

frost toe*

If the aasa of the structure were very large relatively and t In

3*3.1 j&SHK. 1 teirsAs daq StiL Ereat Toe .. tb urer Toe Sot lift**
Brow Fig. 3-Cat

3^ 9- Be

3 = ese

y =• + T2.Q

X = QD

Boar small value, of e

rn t * &• %4 ««

<Xi <49 <«MI <w4 Jj

J XT •

/

/ [ ~ l' ~ v£

^- j .: =

c^jt

IH =0

33 .

= O

R H - rr> X = P x - rnD 6

£V = O

- ^ vc -V mij 4 Kv ^ -V

f?vc = I g 2L - mB6 - SBKv6
2~ JV1 = 0 about cente»- of mass

PhD + q vc ^- Kv 3 B-^|^ - (me

Substf tatfng -for (^ H cmd £? vc

(d'+bV^) + 4 B* Kv 8 = pD

but (D>B%() = ^

.-. § + 4 ff^e = -^-

m ^ m

■fro m lo b f c W

Since at t = o, ©^B = o

.A

too ul'l coooc; to touch pacing c tAwi ** 0# cr*

■*k

m B 0 + 2 BK V 0 * - r ^ L

rs*o» vMch

(>i }

e = c, cos t + -43 . r „ TTKr,

^ 4£K V + C*. S»n|— m t

0 $ 0 = O qt t - O

C.~ -*?3 .

' 4 6 /r,

Cz - o

0 =H 1

'jlV - ?■ = A ^-7 '

r 2>

/' n f - f ~ v/1 4- f’V'

N t-F- ^

^ Jr

•f~ A "

V'l

V

• •

<.111

4 v *,i 3--> - H “•"s'

w/'

fc’l- i'r? ir, ijlits-

p r. i\i “I

ls-> j _ s| “ ./ _ /-

+ a

3V ^ 1

M A

_ v W bn t //I '°t t ,,lt> ‘ i ’ l ' d ^

i'r/ = -4. (Avvi+*a; 3n-/

- v-v'i-A'i)

JT -

1.

I / U

- j I'J- S± ^ -r e /.

) . !

41 4

A -;~ **v

„ - ^ , 0 ; - J ;

J r 1 ■* 9

an -c

' , -t F T-

Y/> - J t

.? -J ,

I '

- i j-jc

aa - « a$
s

a SI

M n 1

r r--v

, >* /} ^ ! ~

f — ' | r/ic
J rl •

v ^ .U t

- 4

. jJLl s

r-ri '

^ = 1

/*■ — -< *-} *~j

S' lj ~t S

sytA _

1^3

O -

= s>

34 .

Artl

e « m3

^-BKv

( l - cos'!'

ZKv

m

t )

or Sfnce V=

$ = cos ]d "s' t)

oil)

rut froH equation (3-11)

_/RD

4£>*K>

If l -co^

t >

0^ Sfnce K v - — -^-

2 S

0=C^n 3 X9 B )HX3X »- »*•

ttf£i

)

V K< $

F crusting the Velma of & t

ted eolvin/ for p/ag —

R

\ - cosICSfj) t t

012 )

Equation 012) dv«e the tta© which a given pulse* F-, * sust act
on • rigid siaucture having a given ng awl V® bofore tho roar toe ul 1
lift.

If the pulse acts « shorter tine thee that of equation (3-12) the

rear too will not lift. The instant the pulse rtoj the eceelaration of
the nsoa vertically downward will te a* a deceleration and the inertia

fY I u

[ 2 C 0

) :

■: i

r

v

a i

pi <i

v/if

- - 4

v*

' 3
jl, l\l

2

i * •* ^ \ r

l « Ja tTJ --- r -

ir,

■ t .-; /i *-j¥v I ..-o-

fl 9

*

r t
i

Vi ^ . j, ' ' • u. *

.!•■•' -f k TV-'

, Y — ~Ji.

v / 1 »- r

r r* —

V-Y ?

<• ‘X N 4 ■ '

= J

32ni o <o

P ^_i . * , - <J x » *■ - , r

. / : ^ Z T«‘

c * f

4 > v V '

1 A 5 i r *V/

J V j >.^/ i

4 j. L -

•' O i k~s~Z) ‘

.r o- y

j — . j

/

A~ ,

'.y

t t

~rrT

35

force will *ct downward instead of upward. This downward foroo plus the
weight of the structure will keep the rear too down.

Relation (3-12) therefore represents the lainteuw coohinatimr cf
pulse aegnitttde sal duration which will cwus? the rear toe to lift.

In appendix C, Fig. C-6 la a plot of ecuation (3-12) for P/D » .
aad §/S « 25. The ratio F/D gives the gecaaotric jrepertione of the
structural the ratio D/S indicates that • , the iraaediate eettlsnent cs the
structure ia placed upon the ground (springs), is 3/25 of P, the height of
the cantor of naas.

th. «em. ^ uhich tto rtructow to s mri « tto n« to.
lifts, is iven by substitatlng values satisfying equation (3-12) in
equation (3—11). The apprco&wsta depression of the spring under too A
would be g - g B 0 f * *^ b9 war*' done on the spring (ground) would than be a

V = iKvy> T*fv{4£.X) = SKvB'e’

013)

3*3*2 IteRlI* to

ftwt Too

As discussed previously, this front epring, A, rwy deprwc* More
after the peer toe lifts. For cxanpla, if tie ossa were large and the
spring soft, this additional depression of the spring A caused ty an
•ajltloMl rototl* e 2 «toU to tfplltonl. On tto other torn. If tto
mum were relatively swell and the »prtng relatively stiff, the spring

would depress only a ir tly the c«xb» would rise sa the structure :vtst~
ed. lth the spring characteristic shown in Fig. 2-4, the spring A will
retain its position of modUaun depression.

’ J .

•ft

0 *i z = u ■ 4

1 J ^ i u

C -T A # f > -

\a j\ ;i S - l ,<3 i^)v/ f ? " ( L >■'■• -

,n J - V

36

The deten ^notion of 6 la complicated by the fact that veil-

T3SK

ntionc In the duration of tbo puls® chenge the fora of tbs calculations.

In addition the rotation is taking place about a point (toe A) which la
nerving vertically. The reversal of inertia forces can causa the rear toe
to <£>ao back down temporarily on ejaring C.

To find the value of t» 2 for a given case it Is necessary first to
determine free; aquation (3-12) or frara a graph such as Fig. C-6 wLatfaer the
in ;ulce is sufficient to cocoa lifting of the roar too. It is than
aaaaaaary to write the differential equations of Motion Involving $ Md y
and solve for 6, sad t for the different stages of the action, i.e.
lifting of rear toe, ending of pulse, and point where rotation stops.

To perfera the above calculations is tedious, end no staple
general equations can be written. For a specific problem, a solution can
be obtained very nicely by numerical Methods such so aro discuacod in ref-
erence 5, p. 50. Solution ty numerical aetljoda baa tl» added virtuo of
giving a step-ty-etep picturo of the Motion of the structure. However, the
result will not be ary bettor than the amnjgptices Bade to a tart with, and
the labor of solution by either analytical or msrsorical Methods nay bo un-
warranted.

3 * 3*3 • jjctgfrioQ If anti & EtruofogT? sq ia.*dd Cround

In lieu of cither cf the calculations just described, bom
qualitative lnforaation can ba obtained by eon, airing enercy levels. To
do this it la necessary agoin to Maka the aaru { ticn that tbs initial

rntua of the structure is octal to the isipulss of the load* The initial

kinetic energy can tlao be found, and this initial kinetic encr is the
total energy of the structure at aero tire# fio acre energy Is added to the
sysrten. The initial K. • is used up ty doing work on the structure*

Since the structure is rigid, all the vork dona is external work. inee it
has been assisted that the borlscctal ground resistance is lar f 7e enough to
jswrent sliding, the external work consists of rotating the structure*

fig* 3-9 shows identical rigid struotueue placed (a) on yield-
ing ground and (b) on unyielding grot • oth structures hose road x. the
aama WDdnm angular displace > <rt 1 . * but tbs cue in (a) has alao do-

BcK

prmioed the spring under the front too ^distance of f L * The weak roquirea
to i*t these structures in thass positions will be found.

thought of in two paste; first, the &epr« sing of the front spring a dis-
tance y^ as the structure rotates an amount ¥j$ second, the rotation fro*
with yj constant* The expression for work isj

e,

M is the accent recuirod to rotate the structure end is a function of *».
Hit

The work to put structure (e) in the position shown can bo

•7 X) ,V? /

i. - |

+ ,6 v 1

2 ioW

r y .

Pf

^ it

^ v /I

> , ,3 tl

. ^ r~r \
>1 * —

/ X'f

r M

■ e a

^t.M 1/ + Va

o'

i *

- ( /i 't oW

rw

t# oSt

(b) ir th» shMB i* :

r 0 max

Wo r K . = j \v\d0

b o

•orJLj bn ur J t"*

r 0 \ r Brnxx

Wot< , = j \V\dQ 4 i \V\ c *9

fc ° 0 ,

^ £& * SjJl t he fiKTMRt

weld

M = (B - "OB')

t^T this v» turn 1 * \tr t r-» t ter- —

^ 0 , /- 0 m ax

VJorK - ^ (grDBjd© -v V\ d0

o v Je (

- m 3 ( 50, - ) 4 > ^ ia V\di©

cw essaae t*t thess two quant Itien of wop
th»t e gives ir: tUl . . would sreduoe the u.^m

(•) »o tb,.

ju

jWorK^ = WorK b __

Wg lSl f 0 wia* _ -j. r B may:

~ B e » 4 wi de ~ mg(Be x - ^| j- 3 v j_ wide

1 6 1

ft B = ^ (

)

^ i }

crtT 35 H p /si & /pf + !/£

ti*S 3 ( 3 - 14 . J* # bftt» !*«'» t' fi fW U ' »<_• **

1 * .ttlao, <*nf . - involve* the riwH ri -f U» *

^ ti-i i>-xi ton first 'fVi.

*4

*r ;

M

> S cd -

? c M

, /i ■* c'W

Cl

/; i <v 9
9 / /V'

) . ~ ■)
- f- d L hi

a r-'

J

)•< 1 f'\A/

a

> i - h) d, y - n

V ~)

/ _ r -U’^a-s? c ri y - >ioV

'-'J « - rd

i-ti-'

I v -
M

^ ^ rr

/- 4 a*

/*icV/ - }S '/I -to'\V

.c _

t ^ 3 7

, ^
^ vfj U

8 ™

■* f > »V 7 ^ *■ 7 , •

' — . _ ^

v A 1 + "V v3 , W ' x

,6 i i'i;

39

TLX* ham beam .-r^ICKjaly dealgneted

If, far « <4««t iMi applied to « jivae »tr» it~*^ on yi* 1-3 • d

, t

of * .’ te (OVl iWi £v m %h* r 1 ! lilioft * y^/ — • 1 Ji ^ e * 1 to

oorpar* ' It?, th « U »1 £, ^*n 4 rrv .- . or • - i 0 - 14 ). - ^

mw,<H»<a tLr ft'rvi'i e«e i- i.fer--.- *

IX 4^ < crit ttar atrttirtare will been • lower aa arfr—
rotation «o the yl« ' ii . c*-> » • io on ar^rialdiri- growl.

If » erli #j, the straotisr will l the aon Mrf. m
rei»*j,® os elthsr tW» yi.»l *ro»& cr ^ ’cuteMlnr g tt a ao l .

If ^ crit £j* the ehmeta^ will bew « y after bbUb
rotation on the yl*15l ng grand than on *.!♦' ^rtnajd.

T 1 *> tbIob of i ia not tacrx. exactly* St 1- r *1 to •• ( v . tor
than which ia ft «uaglr art which the r©«r toe llfte. H» ou» x
•• ta found from vuaticoe —12) and (-*11).

■fence #j * #j,t it foUnan that > crit ^ tfen Knafent

will rotate fart:- - rm yietUf; * gnxmd than on nec-yi* Iding '■■*-- • if
» crit £ no definite otutat »*t can be -

value *f critical ^ efetei***' frcaa t-ruation (3-14) am - - a
a H» efew^e of t> •trucUra tl<* flexibility cf the dreaeu .

*

J

t

> :

>

'V.

U a

it «■

Ua « cri CT9> • v ti« 044) j*t« U>r sotil JlmUy

rid-* T* - .

F»*3-l

Force: Diagram

Coordinates Referreo to c.m.

Fig3-E

43 .

(a.)

Flc x 1 ece Structure

Fiq 3-5

Horiz

Force.

k

HoRlZONTAcDlSPI*.

Horiz
Forcs .

mgf

Horizontal Displ,

Resistance of Structure
to Horizontal.
Deformation

Fici 3-6

Resistance of Ground
to Seioinc-i of Structure
a Distance x

Fk-,3-7

u.

IV. . - *Xfl -=1> C'XSkL».H -

dMV'Uwlatlc of >urwr«*v* (2) thoew charset riuti «f tin !***’■ ,)

i* ■ . » mtutf^ir boat, first * ^ '. terad r - •»» " J - *.

flcxi*.:* . T»» luaO'Jor 3* • * .mro lapul** funeti m, sil;, */r 1 7

m ■> * v v «-rvc faros, «*l,t«* *t guil t. s» '*» Jte ■*■*" • "

-^o i# t •-*: -*► • ^ horiaonUl jr©f«rti*» » «ai»t«veo sad i»'i-

csl arU** of rv*djrue»«. |«ib rigid -i» fieri' ’* Tts a*.- ar# eour-

*SPIW* a

i IcUob# for trm. -turning: and » IX -J- ■ ; -M in* 4 for ’ Wic

rigid structure on rlgfc • ■ nd with U» «oJti -ut *swnrfcr«tn' «*•
a^vll-*' at tho c-u.^v Us Wucture* ” :u il afTac** of . 1^,-1 ta

4 .

*t> • f trieio* on ri^ ground i» atruol: b m t*i^- «-va

load nt its

uWU* or

.w

to *

• A#

•d

IJb

W ^ only • both » 2103 i>£

HgM

by

OB

IVr •

tbs

t» « ri<U
the . mart l 1 oi of

^rtanslr*
at the

nireriora im sag!*,

a struct^** ulU slice when struck by «

W *»f borl? gpotai reel

1 t}. t*« . «d <e«l* Of GT5rtum-Sg

cy the *cdai on the structure at stilish ue ecuiirelmst con—
*~vul 0 « U«*. li . ,.:•■* At ’4»* a** . do -ri*--*n
ajlo + wa o s Is sos&ad to pwi .L!<tiu» . A- cm * ^ c. *>
ac In ti*-> emm r t~>- tlou as <.» ooc^ra- ly bo lev. YN
of rotation lw^*, •« V%* of «. lie on of is

T» location -yf ".U. of •o-=aelon it

#vnfc tr* *s*<"rtIo of f It io

of -jtA lw*5 si'" - r*Toti~ to -pc tsl 1 *«
no* 11* witih the } rct*% ^ "«i«a ?f the

•fiVct r>f f U-±i»: 1 ! ty %!•>' • U m etnas **»"* Is to »•»»•« is*
V ellillas trem tl*l «cynl -„■» « r' J - - stanircar .

&U 4 li * th?» iT ^ta* tiiv s -i t m< of *110:14 Alas la Ur-

of tba load --M. . wow*!?, cv skdlltr to*

4s! .

jjrtotieoe

thacla

«# filler i

X

s

9

D

t

**
K-' .

If

'l

S

s

Si

St

-f toe 1 of Hn^a» i<f > W to iidti*? Mit

• fmXtlv to ri/*t, y pcsitina

«... «t«l u*e4 in aoctl*.r ?• *3*

half wi 'th of structure ,

hei^t of c« • / < atroritara la i» fxdtlol r»«t ^e'tiaa*

of •t ruotur a it* c*.

to hflriWWl lOttl.

#i*>ing

I*

5

4

4 "*.

(<

)

t

wmi -r n)-* t wk Ich i 1 Lolnc «&3y.

dur*(.l.«« «/ i,.

■ w 1 1 v*Atnm of an+tlaa * iWa fciw it w l it* c«au

.Jk

»wvV^. :•/ w .7*»ti0» of struetux* «bout too A.

50 ,

1 ,

* * I

"ftrtursl Ifcriou' of ifaifora Osntilamsr i
Jl* Proowofif 1 , i 1‘ .s' .

2, Jeoob&eu, ,iUt - .« "Lynardc Dehtndcr of wi v^J fled rtructurea u*

to the feint of Collapse*® OCUL Corferenou,
Juae 1952*

3* feumrfe, • «s •"ethous of Impels for taructuree «Aii.;)cctedl

to >maRlo loading.* Directorate of
IfetelHcaooc, CUF, 1 50 .

4, .. . t 1 i*atyi&8 end Deaton of Structures i Jocted

to l£m*edc loading.® i<l? Cooferono on till-
ing in tbo itonie igu, 15 Juno 1952.

5 * newaark, ». m .cogatetiofci of J%nrwric truetsral Keeper*# is

the l^cgo j^jraaohix Ft&lsan." Cer~

forerxr, Juno 1 '2.

6 .

‘ Oil ‘Oft- ^

ofon <■**: ~**9

■lamtiim i^oygas 1- ’ 'rri&> a»ing . w
u, Vastr*** -oc.. «t$r, Ine.t ‘»w Ibrl*, 1*77.

Feck, ♦Ifij ».t
7. Tinoeb nko, &«•

51

»

U

u.

53

o

to

in

oi

O

f

vJ

in

O

in

c>

o

10

OMJ

or* u.

54

55

X

Value.* op Puu&i Ma«»nituo« amo Duration Which Will
Cau*c. Rear. Toe to L\pt for a RiCjio
Structure on Yiewoincj Quouno

56

in

oJ

0.3 0.4 0.5 0.6

F i C“ 6

57 .

Thesis

E155

APR

FEB

MAY
to V 2 I 56
!•! V 3 1 61

2
I 0

0 IHDERY
n c c A T

-**’ 1 0 3 9
0 I SPLAY

10 3 9

10 3 9

23004

Thesis

3165

Hammer

The influence of founda-
tion coupling on the dyna-
mic response of simple

structures .
-%

B I tlDERY

1 1 - I

FEB 2
MAY 1 8

Ml 2 1 56

HE CAT

D I SPL/^

10 3 9

. Y 3 I 6 1

' r * n

23004

Hammer

The influence of founrt*ti«n
coupling on the dynamic response
of simple structures.

library

u. S. Nava Postgraduate School
Monterey, California